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Full text of "Analytical Mechanics"

154 ANALYTICAL MECHANICS PROBLEMS. 1. Find the moment of inertia of a circular lamina about a diameter. 2. Find the moment of inertia of an elliptical lamina about its minor axis. 3. Find the moment of inertia of a rectangular plate of negligible thickness about a diagonal. 4. Find the moment of inertia of a thin plate, which is in the shape of an equilateral triangle, with respect to one of its edges. 5. Find the moment of inertia of a triangular plate about an axis which passes through one of its vertices and is parallel to the base. 6. Find the moments of inertia of the following laminae with respect to the axes indicated by the thin vertical lines. a 2a ?hg 2b 2a III (a) (c) (d) (e) 121. Theorems on Moments of Inertia. Theorem I. — The 'moment of inertia of a lamina about an axis which is perpendicular to its plane equals the sum of the moments of inertia with respect to two rectangular axes which lie in the plane of the lamina with their origin on the first axis. , Suppose the lamina to be in the zy-plane, then the theorem states that the moment of inertia about the 2-axis equals the sum of the moments of inertia about the other two axes, that is, I. -!,+ !,. (Ill) The following analysis explains itself. Xm r2dW? / (£fit/ = r\x* + y*)d JO rf* x2dm + I Jo